David E. Rowe,
Department
of Physics, Mathematics and Computer Science, Mainz University:
On
Proofs of the Transcendence of e
and π as Background to Hilbert's
Seventh Paris Problem

e^{πi} + 1 = 0

Hilbert’s
seventh
Paris
problem
concerns proving conjectures about the
irrationality or transcendence of certain types of numbers. The
inspiration for such conjectures came from two famous results obtained
earlier: Hermite’s proof of the transcendence of e in 1873 and its extension in 1882
by Lindemann to the case of π , which he obtained by showing that
the equation e^{z}+1=0
has
no algebraic solutions.
In 1893 Hilbert managed to prove both results in just four pages, a
feat that led to further simplifications by Hurwitz and Gordan. The
following year, Klein taught a course for future Gymnasium teachers in
which he gave a detailed elementary proof published in 1895 and in
numerous translations thereafter. After this, few people probably ever
again read Lindemann’s original paper, while the curious and rather
amusing story behind it went untold. Soon after it appeared, Lindemann
became famous for having resolved (in the negative sense) the ancient
problem of squaring the circle, which Greek geometers had only been
able to solve by means of transcendental curves. After describing this
classical background, we turn to the events of the 1880s and 90s, the
entangled careers of Klein, Lindemann, and Hilbert, and to a
reconsideration of these events in the light of Hilbert’s seventh
problem and its place in the history of mathematics.
tirsdag, den 8.
marts 2011,
kl. 17.00


